# Comparable Dose - Haber's Law

What is Habers Law and how is it applied in inhalation toxicology?

Habers Law or Habers Rule, as used by inhalation toxicologists, states that identical product concentration (c) of an agent in air when multiplied by the duration of exposure (t) will yield an identical biological response or a constant. Empirically this may be expressed as (c)(t) = constant.

This was developed by a German chemist Fritz Haber (1868-1934) to characterize acute toxicity of chemicals. This simple equation was used as a scientific basis for the setting of exposure limits. However, animal experiments demonstrate that this rule does not always apply. In fact, whether a study confirms or refutes the law may depend on the endpoint of study or the toxicological response.

For example, phosgene's effects on pulmonary gas exchange and magnitude of pulmonary performance were constant for a variety of concentrations and exposure times. In this case, the rule fit the data. However, when the effects of subchronic phosgene exposure were investigated, the data revealed that the phosgene concentration rather than the (c)(t) product was more closely related to the the toxic response.

This apparent contradiction also holds true for ozone. Studies using the identical exposure protocol, but different toxicological endpoints (protein from bronchiolar lavage versus cell renewal in the airways) reveal that the driving force is the concentration rather than the (c)(t) product for the latter while the rule is valid for the former toxicological endpoint.

It is generally agreed upon in the inhalation toxicology community that Habers rule does not apply to all toxicological endpoints, and even where it does apply, it is only valid over a finite range of exposure concentration and times. Thus, dose and the effect produced by that dose, as the most important variable in toxicology, from Paracelsus to the present time, still reigns.

For inhalation toxicology, concentration of an airborne agent, which eventually will determine the dose, is generally considered the most important determining factor in toxicity.

**Reference: **Some Notes on the History of Haber's Law. Witschi, H.,
Toxicol. Sci. 50:164-168 (1999).

**By: Arlene L. Weiss, M.S., D.A.B.T**

**Postscript:**

Some toxic chemicals have critical levels of toxic mass (m in grams,g) per body weight (M in kilograms, kg), which is a mass fraction measure of concentration in the body. If there is no loss (excretion or destruction), then this measure is proportion to the product of the concentration, the time duration, and the inhalation rate, for a particular animal. Thus follows Habers rule as it relates to delivered dose.

If there is a loss, then the g/kg fraction will depend on the rates of input and loss. The following equation describes a simple model for the rate of change of the mass fraction (m/M):

(dm/dt)/M = (Q c - L m)/M

where

Q = breathing rate, c = toxic concentration in air, L = loss rate per mass of toxic material in the body, and t = time.

This equation gives an approach to equilibrium having its time dependence proportional to [1-exp(-t/t*)], an asymptotic approach, and the equilibrium value is

(m/M)* = Q c / M L ,

showing the influence of breathing rate, concentration, body mass, and loss rate. For times short compared to t*, (m/M) = (t/t*)(m/M)*, a value is generated that is proportional to (c)(t) as in Habers rule.

**By: Douglas W. Cooper, Ph.D.**